There are two different formulations of the rigid connector, rigid and flexible. In the flexible form, the rigid motion is prescribed only in an average sense, and instead an assumption about linearly distributed traction fields is used.

The only degrees of freedom needed to represent this assembly are the ones needed to represent the movement of a rigid body. In 2D this is simply two in-plane translations, and the rotation around the z-axis.

In 3D the situation is more complex. Six degrees of freedom, usually selected as three translations and three parameters for the rotation, are necessary. For finite rotations, however, any choice of three rotation parameters is singular at some specific set of angles. For this reason, a four-parameter quaternion representation is used for the rotations in COMSOL Multiphysics. Thus, each rigid connector in 3D actually has seven degrees of freedom, three for the translation and four for the rotation. The quaternion parameters are called a, b, c, and d. These four parameters are not independent, so an extra equation stating that the following relation is added:

The connection between the quaternion parameters and a rotation matrix R is

Under pure rotation, a vector from the center of rotation (Xc) of the rigid connector to a point X on the undeformed object is rotated into

where x is the new position of the point originally at X. The displacement is by definition

where I is the unit matrix.

When the center of rotation of the rigid connector also has a translation uc, then the complete expression for the rigid body displacements is

The parameter a can be considered as measuring the rotation, while b, c, and d can be interpreted as the orientation of the rotation vector. For small rotations, this relation simplifies to

The rotation vector is available as the variables thx_tag, thy_tag, and thz_tag. Here tag is the tag of the Rigid Connector node in the Model Builder tree.

Instead of enforcing the rigid body constraint on the selected boundaries in the pointwise form, it is possible to use a weak form of the constraint. This is invoked by selecting Use weak constraints for rigid-flexible connection in the Constraint Settings section for a rigid connector. The weak form is implemented as

Here u is the displacement field on the boundaries, ur is the rigid body displacement as given by Equation 3-110, and Frs is the reaction force field (Lagrange multiplier).

The flexible formulation of the rigid connector is based on the weak expression of the constraint, Equation 3-111. In the original weak form, however, the reaction force field can have any distribution, in order to enforce the rigid body motion.

Here Fc (unit: N) and Fd (unit: N/m) are two global vectors, each having three degrees of freedom in 3D. Fc can be directly interpreted as the reaction force at the center of rotation. Fd can be considered as a representation of the gradient of the reaction force field.

In the case of geometric nonlinearity, Equation 3-112 is replaced by

where R is the rotation matrix. In 2D, Fc is a vector with two components, whereas Fd is a scalar.

If the boundaries selected in the rigid connector are not contiguous, then each set of connected boundaries will have its own set of Fc and Fd degrees of freedom. The center of rotation, Xc, is then taken as the center of gravity for each individual group of boundaries.